Laurent Manivel

On the Ideals of Secant Varieties of Segre Varieties

Abstract We establish basic techniques for determining the ideals of secant varieties of Segre varieties.We solve a conjecture of Garcia, Stillman, and Sturmfels on the generators of the ideal of the first secant variety in the case of three factors and solve the conjecture set-theoretically for an arbitrary number of factors. We determine the low degree components of the ideals of secant varieties of small dimension in a few cases.

Generalizations of Strassen's Equations for Secant Varieties of Segre Varieties

We define many new examples of modules of equations for secant varieties of Segre varieties that generalize Strassens commutation equations (Strassen, 1988). Our modules of equations are obtained by constructing subspaces of matrices from tensors that satisfy various commutation properties.

The Chow ring of the Cayley plane

We give a full description of the Chow ring of the complex Cayley plane, the simplest of the exceptional flag varieties. We describe explicitely the most interesting of its Schubert varieties and compute their intersection products. Translating our results in the Borel presentation, i.e. in terms of Weyl group invariants, we are able to compute the degree of the variety of reductions

Representation Theory and Projective Geometry

Y_8

Vanishing theorems for ample vector bundles

introduced in our related preprint math.AG/0306328.

Hypersurfaces with degenerate duals and the Geometric Complexity Theory Program

This article consists of three parts that are largely independent of one another. The first part deals with the projective geometry of homogeneous varieties, in particular their secant and tangential varieties. It culminates with an elementary construction of the compact Hermitian symmetric spaces and the closed orbits in the projectivization of the adjoint representation of a simple Lie algebra. The second part discusses division algebras, triality, Jordan agebras and the Freudenthal magic square. The third part describes work of Deligne and Vogel inspired by knot theory and several perspectives for understanding this work.

Quantum cohomology of minuscule homogeneous spaces III Semi-simplicity and consequences

Summary. The main result of this article is a general vanishing theorem for the cohomology of tensorial representations of an ample vector bundle on a smooth complex projective variety. In particular, we extend classical theorems of Griffiths and Le Potier to the whole Dolbeault cohomology, prove a variant of an uncorrect conjecture of Sommese, and answer a question of Demailly. As an application, we prove conjectures of Debarre and Kim for branched coverings of grassmannians, and extend a well-known Barth–Lefschetz type theorem for branched covers of projective spaces, due to Lazarsfeld. We also obtain new restriction theorems for certain degeneracy loci.

On the period map for prime Fano threefolds of degree 10

We determine set-theoretic defining equations for the variety of hypersurfaces of degree d in an N-dimensional complex vector space that have dual variety of dimension at most k. We apply these equations to the Mulmuley-Sohoni variety, the GL_{n^2} orbit closure of the determinant, showing it is an irreducible component of the variety of hypersurfaces of degree

Quantum cohomology of minuscule homogeneous spaces II : Hidden symmetries

n

Secants of minuscule and cominuscule minimal orbits

in C^{n^2} with dual of dimension at most 2n-2. We establish additional geometric properties of the Mulmuley-Sohoni variety and prove a quadratic lower bound for the determinental border-complexity of the permanent.